Geometry Theorems

Geometry theorems are key concepts that help solve various problems on the PSAT. Understanding these theorems, like the Pythagorean Theorem and properties of triangles, will boost your confidence and skills in tackling geometry questions effectively.

1. Pythagorean Theorem

   • States that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): ( c^2 = a^2 + b^2 ).
   • Used to determine the length of a side in a right triangle when the lengths of the other two sides are known.
   • Essential for solving problems involving right triangles in various geometric contexts.

2. Triangle Sum Theorem

   • States that the sum of the interior angles of a triangle is always 180 degrees.
   • Can be used to find missing angle measures when two angles are known.
   • Important for understanding the properties of triangles and their relationships.

3. Parallel Lines and Transversals Theorem

   • When a transversal crosses parallel lines, several angle relationships are formed, including corresponding angles, alternate interior angles, and consecutive interior angles.
   • Corresponding angles are equal, and alternate interior angles are equal, which can be used to prove lines are parallel.
   • Key for solving problems involving angle measures and parallel lines.

4. Congruent Triangle Theorems (SSS, SAS, ASA, AAS)

   • SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
   • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
   • ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) also establish congruence based on angle and side relationships.

5. Similar Triangle Theorems (AA, SAS, SSS)

   • AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
   • SAS (Side-Angle-Side): If two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar.
   • SSS (Side-Side-Side): If the sides of one triangle are in proportion to the sides of another triangle, the triangles are similar.

6. Midpoint Theorem

   • States that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
   • Useful for finding lengths and establishing relationships between segments in triangles.
   • Helps in solving problems involving triangle properties and segment relationships.

7. Angle Bisector Theorem

   • States that an angle bisector divides the opposite side into segments that are proportional to the adjacent sides.
   • If a point D lies on side BC of triangle ABC, then ( \frac{BD}{DC} = \frac{AB}{AC} ).
   • Important for solving problems involving angle bisectors and segment ratios.

8. Inscribed Angle Theorem

   • States that an inscribed angle is half the measure of the intercepted arc.
   • If an angle is inscribed in a circle, the measure of the angle is equal to half the measure of the arc it intercepts.
   • Useful for solving problems involving circles and angles.

9. Tangent-Secant Theorem

   • States that if a tangent and a secant are drawn from a point outside a circle, the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part.
   • Formula: ( (tangent)^2 = (secant)(external , part) ).
   • Important for solving problems involving circles and tangent-secant relationships.

10. Area and Volume Formulas for Common Shapes

    • Area of a rectangle: ( A = length \times width ).
    • Area of a triangle: ( A = \frac{1}{2} \times base \times height ).
    • Volume of a rectangular prism: ( V = length \times width \times height ).
    • Volume of a cylinder: ( V = \pi r^2 h ).
    • Area of a circle: ( A = \pi r^2 ) and circumference: ( C = 2\pi r ).